nLab van der Blij lemma

Idea

van der Blij’s lemma is a fundamental insight about the connections between characteristic elements of a symmetric unimodular bilinear form and its signature.

If the manifold is additionally smoothable and closed, van der Blij lemma improves to the more well-known Rokhlin theorem, which features another factor of $2$ corresponding to the Kirby-Siebenmann invariant

Statement

Proposition

For a symmetric unimodular bilinear form $Q\colon\mathbb{Z}^n\times\mathbb{Z} ^n\rightarrow\mathbb{Z}$, a characteristic element $c\in\mathbb{Z}^n$ (fulfilling $Q(x,x) mod 2=Q(c,x) mod 2$ for all $x\in\mathbb{Z}^n$) fulfills:

$$Q(c,c) =\sigma(Q)mod 8.$$

Proof

Let $c,d\in\mathbb{Z}^n$ be characteristic elements, then there exists an element $s\in\mathbb{Z}^n$ with $c=d+2s$, which causes:

$$Q(c,c) =Q(d,d) +4(Q(d,s)+Q(s,s)).$$

Since $d\in\mathbb{Z}^n$ is characteristic, $Q(d,s)+Q(s,s)\in\mathbb{Z}$ is even, which causes $Q(c,c)=Q(d,d) mod 8$. Without loss of generality, let $Q$ be odd and indefinite. (Otherwise consider the odd and indefinite form $Q'=Q\oplus[+1]\oplus[-1]$ on $\mathbb{Z}^{n+2}$ with characteristic elements $(c,\pm 1,\pm 1)\in\mathbb{Z}^{n+2}$ for every characteristic element $c\in\mathbb{Z}^n$ of $Q$ and $\sigma(Q')=\sigma(Q)$.) According to Serre’s classification theorem, $Q\cong m[+1]\oplus n[-1]$ for some $m,n\in\mathbb{N}$. With characteristic elements $c=(\pm 1,\ldots,\pm 1)\in\mathbb{Z}^n$, one has $Q(c,c)=m-n$ and $\sigma(Q)=m-n$.

(Scorpan 05, p. 170)

If the form is even, then $0$ is a characteristic element, which yields:

Proposition

An even symmetric unimodular bilinear form $Q$ fulfills:

$$\sigma(Q) =0 mod 8.$$

If the intersection form of a topological 4-manifold is considered, then the manifold being spin implies that its intersection form is even according to Wu's formula. If the manifold is simply connected, then the reverse holds:

Proposition

The signature of a spin topological 4-manifold fulfills:

$$\sigma(Q) =0 mod 8.$$

If the manifold is additionally smooth and closed, then Rokhlin's theorem even assures $\sigma(Q)=0 mod 16$.

Related entries

Articles on geometry and topology of 4-manifolds:

• Basic concepts:

  • Donaldson theory and related articles
  • Seiberg-Witten theory and related articles
  • intersection form
  • Kirby-Siebenmann invariant
  • signature

• Important examples:

  • Barlow surface$\;\mathbb{C}P^2\# 8\overline{\mathbb{C}P^2}$
  • Dolgachev surface$\;\mathbb{C}P^2\# 9\overline{\mathbb{C}P^2}$
  • E₈ manifold$\;M_{E_8}$
  • exotic $\mathbb{R}^n$
  • fake second complex projective space$\mathbb{C}P^2$

• Central results:

  • Donaldson theorem (Fields Medal 1986)
  • Freedman classification (Fields Medal 1986)
  • Furuta theorem (10/8 theorem)
  • Hirzebruch signature theorem
  • Hitchin-Thorpe inequality
  • Quinn theorem
  • Rokhlin theorem
  • van der Blij lemma
  • Wall theorems
  • Wu formula

• Open problems:

  • 3/2 conjecture
  • 11/8 conjecture
  • simple type conjecture
  • Poincaré conjecture

References

• Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society (2005) [ISBN 978-1470468613]